Interpretation of coefficients

[Lanna Ng]

Hello,

I have an experiment design like this:
- At the village level: 20 people receive treatment 1
- Within the 20 people, I randomly selected 10 to give them treatment 2

I estimate the treatment effect using this formula:
Y= a*T1i + b*T2ij
T1i: Treatment 1 at village i
T2ij: treatment 2 of individual j in village i

Obviously those who receive T2 also get T1.

In my data, I have two variables treat1 and treat2. treat1=1 for all receiving T1, and 0 otherwise, and treat2=1 for all receiving treatment 2 (and for these observations, their treat1=1 as well).

My question is, how can I interpret a and b. Is it true that a is the treatment effect of treatment 1 compared to the control, and b is the treatment effect of treatment 2 compared to the control. Or a is the treatment effect of treatment 1 and treatment 2 altogther, because half of the group who receive T1 also receive T2?

Thank you for your help.

[Clyde Schechter]

You can work this out with a little algebra. First, your equation is incomplete, as it lacks a constant term.

For those who receive neither treatment, their expected outcome is E(Y|no treatment) = cons + a*0 + b*0 = cons
For those who receive treatment 1 only, their expected outcome is E(Y|T1 only) = cons + a*1 + b*0 = a + cons.
For those who receive treatment 2 and treatment 1, the expected outcome is E(Y|T1 & T2) = cons + a*1 + b*1 = a + b + cons.

Subtract the first equation from the second to get:
E(Y|T1 only) - E(Y|no treatment) = a.

Subtract the second equation from the third to get:
E(Y|T1 & T2) - E(Y|T1 only) = b.

So a is the difference between Y with T1 and Y from no treatment And b is the difference between T1&T2 and T1 only. The difference between T2 and no treatment is not estimable in this study.

[Lanna Ng]

Thank you. It is super clear and very simple to look at things.